Canonical Thermodynamics

Abstract: In the paper, thermodynamics of canonical systems is derived from the multinomial distribution of the occupancy numbers of quantum eigenstates. The cathegorical distribution (i.e. the one-particle distribution) on which the multinomial distribution is based, should be derived from the maximum entropy principle, but, being the multinomial distribution intractable, the paper proposes to take instead the Boltzmann distribution as cathegorical distribution, discussing the reason why the entropy of the multinomial-Boltzmann distribution should closely approximate the entropy achieved by the multinomial distribution equipped with the entropy-maximizing cathegorical distribution. After this, the imposition of Clausius' equation on the Shannon entropy of the multinomial-Boltzmann distribution leads to an unexpected result: in general, the Lagrange multiplier $\beta$ that imposes the energy constraint in constrained entropy maximization turns out to be substantially different from the inverse temperature. To support this unexpected result, the paper presents an example where, with $\beta$ equal to the inverse temperature, the thermodynamic entropy (i.e. the entropy at a given temperature) of the multinomial-Boltzmann distribution is greater than the Bose-Einstein thermodynamic entropy. However, the latter is derived from entropy maximization with constrained expected energy and expected number of particles, therefore if we plug $\beta$ equal to the inverse temperature in the multinomial-Boltzmann distribution, then the thermodynamics of the canonical system obtained in this way turns out to be non-compatible with the principle of maximum entropy.